Localization effect for Dirichlet eigenfunctions in thin non-smooth domains

Authors:
S. A. Nazarov, E. Pérez and J. Taskinen

Journal:
Trans. Amer. Math. Soc. **368** (2016), 4787-4829

MSC (2010):
Primary 35P20; Secondary 35B40, 35J05

Published electronically:
April 20, 2015

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the localization effect for the eigenfunctions of the

Laplace-Dirichlet problem in a thin three-dimensional plate with curved non-smooth bases. We show that the eigenfunctions are localized at the thickest region, or the longest traverse axis, of the plate and that the magnitude of the eigenfunctions decays exponentially as a function of the distance to this axis. We consider some extensions like mixed boundary value problems in thin domains. The obtained asymptotic formulas for eigenfunctions prove the existence of gaps in the essential spectrum of the Dirichlet Laplacian in an unbounded double-periodic curved piecewise smooth thin layer.

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Additional Information

**S. A. Nazarov**

Affiliation:
St. Petersburg State University, Mathematics and Mechanics Faculty, Laboratory Mechanics of Composites, Universitetsky prospekt, 28, Peterhof, 198504 St. Petersburg, Russia – and – St Petersburg State Polytechnical University, Laboratory of Mechanics of New Nanomaterials, Polytekhnicheskaya ul, 29, 195251 St. Petersburg, Russia

Email:
srgnazarov@yahoo.co.uk

**E. Pérez**

Affiliation:
Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Avenida de las Castros s/n, 39005 Santander, Spain

Email:
meperez@unican.es

**J. Taskinen**

Affiliation:
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FI-00014 Helsinki, Finland

Email:
jari.taskinen@helsinki.fi

DOI:
http://dx.doi.org/10.1090/tran/6625

Keywords:
Dirichlet problem,
asymptotics of eigenfunctions and eigenvalues,
localization effect,
spectral gaps

Received by editor(s):
May 20, 2014

Published electronically:
April 20, 2015

Additional Notes:
The first author acknowledges the partial support by RFFI, grant 15-01-02175 and the Academy of Finland grant “Spectral analysis of boundary value problems in mathematical physics”

The first and second authors were partially supported by MINECO grant MTM2013-44883-P

The third author was partially supported by the Väisälä Foundation and by the Academy of Finland grant “Functional analysis and applications”.

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© Copyright 2015
American Mathematical Society